The Nine Chapters On The History Of Chinese Mathematics
The Nine Chapters on theHistory of Chinese MathematicsRik Brandenburg Keimpe Nevenzeel†15 July 2007AbstractThis article explores Chinese mathematics from the first archeologicalevidence of numbers on oracle bones (14th century BC) to the time Chinese mathematics became a part universal mathematics (halfway the 19thcentury AD).First a concise overview of Chinese history and in philosophy is given.The ethical oriented Confucianism was the dominant philosophy and consequently little attention was given to the natural world, hindering thedevelopment of natural sciences and mathematics. Due to historical andphilosophical reasons, Chinese mathematics took quite a different paththan its Western counterpart: Chinese mathematics was focused on algebra and practical applications instead of geometry and theoretical reasoning.The Nine Chapters on the Mathematical Art (ca. 1st century AD) isby far the most influential work: it would shape Chinese mathematicsfor centuries to come. Between the 3rd and the 11th century AD, Buddhist and Indian ideas got a firm grip on China, yet curiously Chinesemathematics is barely influenced. In the ‘Chinese Renaissance’ and thesubsequent Mongol occupation between the 12th and 14th century Chinesemathematics will reach its zenith.In the 15th and 16th centuries mathematical development waned andimportant achievements were forgotten. Only after the arrival of European missionary-scientists at the end of the 16th and during the 17thcentury mathematics made progress again. The Opium Wars of the 19thcentury mark the end of the classical China and the indigenous Chinesemathematics would be assimilated by universal mathematics.Two conclusions are reached: (i) war seems to be good for mathematical progress and (ii) from the 15th century on Chinese mathematicspassed away in particular due ot the lack of a general algebraic structure,which prevented structural insight.Keywords: Chinese mathematics, history of mathematics, Nine Chapterson the Mathematical Art.IntroductionChina is one of the oldest civilizations, comparable only to Egypt and Babylonia, which were well versed in mathematics (for a discourse on Babylonian IWI,† IWIUniversity of Groningen- Kapteyn Institute, University of Groningen1
and Egyptian mathematics, see Klassieke algebra door de tijden heen by Vander Elsken, N. and Van der Pol, E.; for an extensive treatment of Egypt, seeWiskunde voor het leven op papyrus beschreven by Jaarsma, K. and Van Oost,K.). But unlike these nations, China developed upon her own until moderntimes.Since it is impossible to Rule an empire as large as or larger than Rome withoutarithmetic and given the millennia that Chinese culture developed relativelyunimpeded, her mathematics achievements are intriguing.In this article the authors aim to share their discoveries about those achievements from the very start of mathematical development until its assimilationby Western mathematics.To do so, we have divided the history of Chinese mathematics into six periods,treating them from chapter 3 to chapter 8: Chapter 3, Prelude ( ca. -200): As will be explained later, not much isknown about the mathematics before the 2nd century BC in China. Thischapter will focus on the things we do know. Chapter 4, Foundations (ca. -200, ca. 600): Central theme of this chapterwill be the most influential book in Chinese mathematical history. Alsosome wonderful achievements during the Chinese Dark Ages will be discussed. This book together with the mathematical developments duringthe Dark Ages would form the foundations of Chinese mathematics. Chapter 5, Buddhism (ca. 600 - ca. 1200): After the Dark Ages a new dynasty took control of China, who reformed the educational system. Meanwhile Buddhistic and Indian influences were at their best. Chapter 6, Zenith (ca. 1200 - ca. 1400): A new war was unleashed, afterwhich China was conquered by the Mongols. In this period of war andbloodshed, ironically, Chinese mathematics flourished more than ever; itsachievements would be the zenith of Chinese mathematics, unparalleledin Chinese history both before and after. When compared to the West,the Chinese where centuries ahead. Chapter 7, Decline (ca. 1400 - ca. 1850): China freed herself from herMongolian invaders, but her mathematics would drop N points. Shewould, however, slowly recover with the aid of Europeans. Chapter 8, Westernization (ca. 1850 - ca. 1920): The end of Chinesemathematics. At gunpoint, Chinese mathematics was assimilated by itsWestern counterpart.2
But before we immerse ourselves in the development of China’s mathematics,we should keep in mind the following quote:Our initial view of the science as a gradual accumulation of isolated discoveries, connected only by their common end is in the lastanalysis teleological. It can point A as a step on the way to B, butit cannot lead to an understanding of how B evolved out of the inner necessity, historical experience, and social consequences of A. A did not appear at a certain time because of some final causeordained its time had come, but rather because it was coherent withother ideas, attitudes, and prejudices of its time.In other words:Science has shaped human history. The advent of agriculture, the mastery ofmetallurgy, the development of a script, all this proto-scientific developmentschanged the course of events. Especially after the Scientific Revolution at theend of the European Renainssance, scientific advancements have played a crucialrole in determining the balance of power; it is above all the high level of scienceand technology that gives the Western world her current dominant position inthe world.But the converse is true as well: the development of science – just as anyother human enterprise – is inextricably linked with the social context in whichthose who develop the science live. The value a society attaches to science andthe manner in which scientists are regarded is of profound influence on scientificdevelopment.Therefore the first two chapters will discuss the political history (chapter 1)and the Chinese philosophy (chapter 2), since these may explain why Chinesemathematics developed the way she did.Our summary and conclusions can be found in chapter 9.3
1The History of ChinaContrary to Western history and philosophy, we expect that most of the readerswill only have rudimentary knowledge of their Chinese counterparts. Therefore,our exploration of the history of Chinese mathematics will start with an overviewof the history and philosophy of China itself.1.1Linear vs. Cyclic historiography: historical patternsWhen history is viewed from the western side of Eurasia, the naturalpattern seems to be for civilizations and empires to rise and wane.By 2000 BC Egypt had eclipsed Sumeria; by 1000 BC Egypt was onthe decline but Babylonia was an impressive power. In time, however, it would be surpassed by the Persians, then the Greeks, then theRomans. (.) Underlying this view of history is an unspoken analogybetween civilizations and human lives or perhaps the lives of competitive individual warrior-heroes. Civilizations have an early, creative,aggressive stage followed by the strong, mature age, but over timelose vigour and become less flexible until eventually they are defeatedin battle or succumb of old age.When history is viewed from the eastern edge of Eurasia, a very different pattern emerges as natural and normal. There is no sense thatyounger civilizations supplant older ones, but that civilization progresses through a series of yin-yang-like reversals of direction fromexcessive disorder to excessive order and back again.(Ebrey, pg.33)Few historians today accept the cyclic view of history, but nevertheless thehistory which will unfold itself in the coming paragraphs shows some strikingpatterns. A large part of Chinese history can indeed be described as a succession of dynasties. In general, a capable military leader unites a divided Chinaunder a strongly centralized civil government, with a scholar-official bureaucracy. Often, China itself is enlarged, because the momentum of the of internalunification victories leads the armies beyond the Chinese borders. “Their successors, however, would not all be supermen able to prevent power struggles atcourt, keep the cost of defense and local administration low, and preserve orenhance sources of revenue, all the while inspiring loyalty through their bearingand virtue. In this view of history, men of ability and integrity – both emperorsand their counselors – could arrest decline or even temporarily reverse it, butinevitably the dynasty would weaken and eventually fall.” (Ebrey, pg. 135.)Consequently, barbarian tribes who are eager to invade the rich China cannotbe hold back any longer. These tribes conquer China and found own dynasty.But because (i) their lack of experience in governing a large state; (ii) theirmuch smaller population size and (iii) their attempts to accommodate the Chinese, Chinese scholars are appointed to organize this new state. With the statebased on Confucian values and the immersion of the foreign elite in the Chineseculture, the barbarian tribes are sinified within a few generations, effectivelymaking the alien dynasty a Chinese one. This dynasty would be strong at first,4
though slowly decline and eventually there would be civil war, till a capablemilitary leader again unites China, etc.So ironically, with the occupation of China the conquerers would not end Chinese civilization, but their own. As a result, Chinese cultural tradition, withits first traces in 8 000 BC, has survived almost ten millennia, although Chinahas been occupied by foreign peoples many times. In the coming paragraphs wewill outline the ten millennia of Chinese civilization, starting by the inventionof the family and ending in the 19th century, when the last dynasty falls.1.2In the beginning.not some Divine Creator, but brilliant human beings where responsible for thesteps that transformed the Chinese from primitive people to a civilization, orso the Chinese creation legends say (Ebrey, pg. 10). The Ox-farmer Fu Xiinvented the family and the domestication of animals, Shen Nong, (whose namemeans literally Divine Farmer) invented agriculture and commerce and the Yellow Emperor expelled barbarian tribes from the Yellow River plane, giving hispeople a place to live. Furthermore, the Yellow Emperor ordered 7 of his subjects to respectively observe the sun, observe the moon, observe the stars, fixthe musical scales, construct a sexagimal system1 , create arithmetic and to usethe previous 6 to develop a calendar (Mikami,pg. 2).2The view that the cosmos came into being on its own, without a Creator toset everything in motion, would become a fundamental difference between theChinese and Western philosophical schools, having major repercussions on thedevelopment of Chinese and Western science, as we will see in chapter 2.But despite a fundamental different cosmological view, it is safe to say thatthe Chinese are among the most ancient civilizations, comparable only to theBabylonians and Egyptians (Mikami, pg. 1).Archeological excavations reveal that around 8000 BC a transition from a huntergatherer to an agricultural society occurred (Gernet, pg. 39), around two thousand years later the agricultural civilization was well entrenched. Stone, boneand (later) ceramic tools were used. The first bronze alloys are dated at 1700– 1600 BC (Ebrey, pg. 25) and gradually the Chinese mastered the techniquesto work with bronze to an unparalleled degree of sophistication (Gernet, pg. 40).The first proof of a written language stems from ‘oracle bones’ used for religious purposes. Most inscriptions are inquiries from the royalty to spirits, as toinform when was a proper time to start a war, the worshipping of ancestors anddivinities, agricultural campaigns, etcetera. The writing was rather complex:on the about 100 000 oracle bones found so far, with in total 5 000 differentcharacters, of which 1 500 have been deciphered. On these bones also the firstnumbers are found, as will be described in further detail in chapter 3. (Gernet,pg. 47 and Schirokauer, pg. 9.)1Anumerical system with 60 numbers in its base.there are Chinese creation myths which stage a Creator who sets everything inmotion. However, these myths are younger than the legends regarding the Yellow Emperordescribed here (Schirokauer, pg. 23).2 Actually,5
The first interesting dynasty is the Zhou dynasty, established around 1050 BC.In the following centuries, Zhou Kings attained larger and larger territory, legitimizing their rule by religion (Ebrey, pg. 31). In absorbed states, relatives of theZhou King or local chiefs were put on the throne as representatives of the King.The local representatives had significant freedom and their own armies; becausetheir positions became hereditary, an influential aristocratic class was created.From 800 BC on, the Zhou Kings could no longer prevent wars between theirlocal domains and incursions from neighboring people (Shirokauer, pg. 25).Although the Zhou dynasty would continue to be for almost 500 more years,their role became ceremonial. On the political side, China proper was throwninto chaos by the continuous power-struggle between the different regions.The ruthlessness of the competition among the regional powers, although uniformly lamented, nevertheless served to foster social, technological, and economical advances. . Those engaged in advisingrulers about state affairs began analyzing basic principles of humansociety and the natural order. Soon the most reflective officials werequestioning established assumptions and values. Their ideas began tobe written down, and the circulation of these treatises further stimulated intellectual debate.(Ebrey, pg. 38.)1.3The age of the philosophersThe era of declining power of the Zhou Kings3 was a bloody one, the regionalleaders all strived for hegemony over the Chinese world. In the struggle for survival, new techniques of warfare developed. The chivalrous, aristocratic type ofwarfare (mainly dependent on the aristocratic military class) vanished with theintroduction of cavalry and (gigantic) infantry armies; at 300 BC states sendout infantry armies of hundreds of thousands of soldiers. Since the victor wasthe leader who could raise the largest army, supply them with the best weaponsand had the most extensive resources, regional rulers experimented with newtypes of government, military strategies and technologies. (Ebrey, pg. 41) Theseleaders, eager to conquer and afraid to be conquered, employed men of ideas ontheir courts to advice them on government affairs, in the process advancing theart of oratory, the development of ethics and the study of logic.The foundations of the Chinese philosophy stem from this era which perhapsprovides an explanation for the focus on ethical issues by Chinese philosophers.Four main schools developed: Confucianism, Taoism, Mohism and Legalism,chapter two will explore their different ideas and the implications for the development of science in more detail. For now, we just note that although the fourschools differ significantly, when compared with the philosophical ideas of othercivilizations it is the similarity that is striking (Ebrey, pg. 58).“Among all Chinese thinkers of the period, order was viewed as inextricablyconnected to rulers, indeed to cosmically based universal kings. It is the universal king who embodies political order . Law, by contrast, was not granted3 Historians often divide the era in two periods: the Spring and Autumn Period (722 BC –403 BC) and the Warring States period (403 BC – 221 BC).6
comparable power by any Chinese thinker. Whether from a Confucian, a Legalist or even a Taoist perspective, law was viewed as an expedient, not assomething noble or inviolable, or something that exists above and beyond theruler. They also shared an underlying assumption that the cosmos came intobeing on its own, without a creator of the sort so important in most Westernthinking.” (Ebrey, pg. 58 and 59.)The power-hungry rulers didn’t only look to each others territories for expansion, the peripheral regions also send expeditionary forces to the ‘barbarious’neighboring tribes. This, combined with increased commercial trade with borderlands, prepared the way for the enormous military expansion that was tofollow after the Zhou dynasty was finally disposed of in 256 BC. (Gernet, pg.73.)1.4The first empires: the Qin and the Han dynastiesBased on Legalist ideals, the Qin state underwent several reorganizations: thearistocratic hierarchical system was replaced by a hierarchy determined by thestate, settlers from other states were offered land and houses, serfdom was abolished and a severe penal code was put in place. (Ebrey, pg. 51.) The power ofthe Qin state rose steadily, till in a series of military victories from 230 – 221 BCQin united China. The leader of Qin took the illustrious title First Emperor anddid not only unify China, he also imposed uniformity on it. The Legalist-basedadministrative system of Qin was extended to all of China, measures of lengthand capacity were unified, standard characters were introduced, etc.But although serfdom was abolished, peasants were still forced to work, notfor someone from the nobility but for the Qin state. The First Emperor initiated numerous public work projects: a network of Imperial roads and irrigationcannels, connecting several parts of the now Great Wall, an Imperial Palace andseveral tombs. (Ebrey, pg. 61 and Gernet, pg. 106.)Criticism was not tolerated and to prevent critics to spread their ideas in writing, all books except for those related to topics like agriculture, medicine ordivination were burned in 213 BC, followed by the execution of 400 scholarswho opposed the Emperor. (Ebrey, pg. 61). Although some books were hiddenand survived, as a result of this book burning our knowledge of the Chinesecivilization and science before 213 BC is limited.However, China was not only unified under one government, it was also unified in its discontent with the unifying government. The continuous militaryoperations, the labor intensive public works, extreme rigor of the Legalist penalsystem (see Chapter 2.4), the impossibility for scholars to express their opinionand the hatred of the replaced aristocracy made the regime very dependent onthe strength of character of the person on the throne (Ebrey, pg. 63). When theFirst Emperor died in 210 BC, the unified China decayed into a several years ofcivil war. In 202 BC, the leader of the Han emerged victorious, crowned himselfemperor and the Han dynasty was founded.Instead of harsh Legalist laws, the Han dynasty used the Confucian notion of7
hierarchy and ethics to control its civil servants. (Ebrey, pg. 79). To realise this,officials had to be trained in Confucian classics and should have a respected character. To train potential officials the Imperial Academy was established around100 BC. This rapidly growing Academy had a few tens of students enrolled atits start, over 30 000 around the second century AD. Since the selection criteriafor government-officials stressed Confucian learning and virtues, Confucianismwould become the most influential philosophy in elite circles. That is not to saythat other philosophies disappeared. First, Han Confucians adapted Confucianism to the needs of the state. (Ebrey, pg. 78) They developed cosmological ideasof the cosmos as being a self-generating and self-sustaining organism. Imperialrule was legitimized by ascribing the emperor with the unique ability to linkheaven, earth and human beings. Natural disasters where seen as signs that theemperor didn’t fulfill its role well.Second, Taoism4 remained of great influence, both at the court and with peasants (Gernet, pg. 160).To ‘accurately’ make prophecies, care was taken to the observation of naturalphenomena. This had a profound influence on the development of calend
nese mathematics became a part universal mathematics (halfway the 19th century AD). First a concise overview of Chinese history and in philosophy is given. The ethical oriented Confucianism was the dominant philosophy and con-sequently little attention was given to the natural world, hindering the developm
May 02, 2018 · D. Program Evaluation ͟The organization has provided a description of the framework for how each program will be evaluated. The framework should include all the elements below: ͟The evaluation methods are cost-effective for the organization ͟Quantitative and qualitative data is being collected (at Basics tier, data collection must have begun)
Silat is a combative art of self-defense and survival rooted from Matay archipelago. It was traced at thé early of Langkasuka Kingdom (2nd century CE) till thé reign of Melaka (Malaysia) Sultanate era (13th century). Silat has now evolved to become part of social culture and tradition with thé appearance of a fine physical and spiritual .
On an exceptional basis, Member States may request UNESCO to provide thé candidates with access to thé platform so they can complète thé form by themselves. Thèse requests must be addressed to esd rize unesco. or by 15 A ril 2021 UNESCO will provide thé nomineewith accessto thé platform via their émail address.
̶The leading indicator of employee engagement is based on the quality of the relationship between employee and supervisor Empower your managers! ̶Help them understand the impact on the organization ̶Share important changes, plan options, tasks, and deadlines ̶Provide key messages and talking points ̶Prepare them to answer employee questions
Dr. Sunita Bharatwal** Dr. Pawan Garga*** Abstract Customer satisfaction is derived from thè functionalities and values, a product or Service can provide. The current study aims to segregate thè dimensions of ordine Service quality and gather insights on its impact on web shopping. The trends of purchases have
Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. Crawford M., Marsh D. The driving force : food in human evolution and the future.
Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.
